I've done some reading on how they cool JWST. It's fascinating and was a massive engineering challenge. Some of thos einstruments need to be cooled to near absolute zero, so much so that it uses liquid helium as a coolant in parts.

Now JWST is at near L2 but it is still in sunlight. It's solar-powered. There are a series of radiating layer to keep heat away from sensitive instruments. Then there's the solar panels themselves.

Obviously an orbital data center wouldn't need some extreme cooling but the key takeaway from me is that the solar panels themselves would shield much of the satellite from direct sunlight, by design.

Absent any external heating, there's only heating from computer chips. Any body in space will radiate away heat. You can make some more effective than others by increasing surface area per unit mass (I assume). Someone else mentioned thermoses as evidence of insulation. There's some truth to that but interestingly most of the heat lost from a thermos is from the same IR radiation that would be emitted by a satellite.

The computer chips used for AI generate significantly more heat than the chips on the JWST. The JWST in total weighs 6.5 tons and uses a mere 2kw of power, which is the same as 3 H100 GPUs under load, each of which will weight what, 1kg?

So in terms of power density you're looking at about 3 orders of magnitude difference. Heating and cooling is going to be a significant part of the total weight.